A real number as a point

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An element of the domain of a real-valued function of a real variable is often called a point. For example, an element (point) ##p## in the domain of a real-valued function ##f## of a real variable where ##f'(p)=0## or ##f'(p)## is undefined is called a critical point of the function. The particular type of critical point ##x## where ##f'(x)=0## is called a stationary point. As another example, "the graph ##y=(x+2)(x-1)^2## cross the x-axis at the points ##x=-2## and ##x=1##." In those cases, why we called a real number as a point? Is it because we view the real numbers as points in the context of the real line?
 

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  • #2
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On the graph I think I should say it cross ##x##-axis at the points ##(-2,0)## and ##(1,0).## I think this is more specfic, but I'm not sure whether your statement is wrong or not.
I think your former example's reason is that the whole ##x-y## plane is the domain of the function. They are also true points, aren't they?
 
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An element of the domain of a real-valued function of a real variable is often called a point. For example, an element (point) ##p## in the domain of a real-valued function ##f## of a real variable where ##f'(p)=0## or ##f'(p)## is undefined is called a critical point of the function. The particular type of critical point ##x## where ##f'(x)=0## is called a stationary point. As another example, "the graph ##y=(x+2)(x-1)^2## cross the x-axis at the points ##x=-2## and ##x=1##." In those cases, why we called a real number as a point? Is it because we view the real numbers as points in the context of the real line?
Each point on a number line represents a real number.

On the graph I think I should say it cross ##x##-axis at the points ##(-2,0)## and ##(1,0).## I think this is more specfic, but I'm not sure whether your statement is wrong or not.
In shuxue's example, it was clearly stated that the points were on the x-axis, so the point on the x-axis where x = -2 is also the ordered pair (-2, 0).
shuxue said:
I think your former example's reason is that the whole ##x-y## plane is the domain of the function.
No, in the first example, it says that the function is of one variable, so the domain is some subset of the real numbers, possibly including the entire real number line.
shuxue said:
They are also true points, aren't they?
???
I don't understand what this refers to.
 

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